# Three-Dimensional Virial Theorem (Quantum Mechanics)

1. Apr 8, 2007

### NeoDevin

1. The problem statement, all variables and given/known data
(a) Prove the three-dimensional virial theorem:

$$2<t> = <r\cdot \nabla V>$$

(for stationary states)

2. Relevant equations

Eq. 3.71 (not sure if this applies to 3 dimensions, but I think so)

\frac{d}{dt}<Q> = \frac{i}{\hbar}<[\hat H, \hat Q]> + \left<\frac{\partial \hat Q}{\partial t}\left> [/tex]

where the last term is the explicit time dependance of the operator Q.

3. The attempt at a solution

Letting $Q = \vec r \cdot \vec p$

$$\frac{\partial \hat Q}{\partial t} = 0$$

and for stationary states:

$$\frac{d}{dt}<Q> = 0$$

so:

$$0 = \frac{i}{\hbar}<[\hat H, \hat Q]> = \frac{i}{\hbar}<[T+V, \vec r \cdot \vec p]>$$

$$= \frac{i}{\hbar}(<T(\vec r \cdot \vec p)>-<\vec r \cdot \vec p T> + <V(\vec r \cdot \vec p)> - <\vec r \cdot \vec p V>)$$

but

$$<\vec r \cdot \vec p V> = <\vec r \cdot (\vec pV)> + <V(\vec r \cdot \vec p)>$$

so

$$0 = \frac{i}{\hbar}(<T(\vec r \cdot \vec p)> - <\vec r \cdot \vec p T> - <\vec r \cdot (\vec p V)>)$$

Last edited: Apr 8, 2007